ON THE RIESZ TRANSFORMS FOR GAUSSIAN MEASURES

Let B be an n x n positive-definite symmetric matrix, and L(B) the sec ond order partial differential operator in R(n) defined by L(B)u = 1/2 DELTA - Bx . delu. The operator L(B) is self-adjoint with respect to t he Gaussian probability measure gamma(n)B(x)dx, where gamma(n)B(x) = C (n, B)exp(-Bx.x). In this paper a class of Riesz's transforms naturall y associated with L(B) is studied. It is shown that these transformati ons are bounded in the spaces L(gammanB)p(R(n)), p > 1, with a constan t independent of the dimension an depending only on p and the number o f different eigenvalues of the matrix B. The proof of this result is a nalytic and uses appropriate square-functions defined in terms of semi groups of operators related to L(B) and the Littlewood-Paley-Stein the ory. The result contains as a particular case some inequalities proved by Meyer and Gundy using probabilistic methods. (C) 1994 Academic Pre ss, Inc.